Let us now see what the two finite universes are like. The difference between them is that in Einstein’s world it is only space that is queer, whereas in De Sitter’s time is queer too. Consequently Einstein’s world is less puzzling, and we will describe it first.
In Einstein’s world, light travels round the whole universe in a time which is supposed to be something like a thousand million years. The odd thing is that all the rays of light which start (say) from the sun will meet again, after their enormous journey, in the place where the sun was when they started. The case is exactly analogous to that of a number of travelers who set out from London to go round the world in great circles, all traveling at the same rate in different aeroplanes. One starts due north, passes the North Pole, then the South Pole, and finally comes home. Another starts due south, reaches the South Pole first and then the North Pole. Another starts westward, but he must not continue to travel due west, because then he would not be traveling on a great circle. Another starts eastward, and so on. They all meet in the antipodes of London, and then they all meet again in London. Now if instead of aeronauts going round the earth you take rays of light going round the universe, the same sort of thing happens: they all meet first at the antipodes of their starting point, and then meet again at their starting point. That means to say that a person who is near the antipodes of the place where the sun was about five hundred million years ago will see what is apparently a body as bright as the sun then was (except for the small amount of light that has been stopped on the way by opaque bodies), and having the same shape and size. And a person who is near where the sun was a thousand million years ago will see what is apparently a body just like what the sun was a thousand million years ago. And the same applies to the antipodes of the sun fifteen hundred million years ago, and to the place of the sun two thousand million years ago, and so on. This series only ends when it carries us back to a time before the sun existed.
But all these suns are only ghosts; that is to say, you could pass through them without experiencing resistance, and they do not exert gravitation. They are, in fact, like images in a mirror: they exist only for the sense of sight, not for any other sense. It is rather disturbing to reflect that, if this theory is true, any number of the objects we see in the heavens may be merely ghosts. They are like ghosts in their habit of revisiting the scenes of their past life. Suppose a star had exploded at a certain place, as stars sometimes will. Every thousand million years its ghost would return to the scene of the disaster and explode again in the same place. There is, however, considerable doubt whether rays of light could perform the journey with sufficient accuracy to produce a clear image. Some would be stopped by matter on the way, some would be turned out of the straight course by passing near heavy bodies, as in the eclipse observations described in [Chapter IX], and for one reason or another their return would not be punctual and exact.
There are various reasons for doubting whether Einstein’s universe can be quite right.[12] Some of these are rather complicated. But there is one objection which is easily appreciated: in Einstein’s theory, absolute space and time re-enter by another door. The ghostly sun is formed in the “place” where it was a thousand million years ago. Both the “place” and the period of time are in a sense absolute. We saw as early as Chapter I that “place” is a vague and popular notion, incapable of scientific precision. It seems hardly worth while to go through such a vast intellectual labor if the errors we set out to correct are to reappear at the end.
De Sitter’s world is even odder than Einstein’s, because time goes mad as well as space. I despair of explaining, in non-mathematical language, the particular form of lunacy with which time is afflicted, but some of its manifestations can be described. An observer in this world, if he observes a number of clocks, each of which is perfectly accurate from its own point of view, will think that distant clocks are going slow as compared with those in his neighborhood. They will seem to go slower and slower, until, at a distance of one quarter of the circumference of the universe, they will seem to have stopped altogether. That region will seem to our observer a sort of lotus land, where nothing is ever done. He will not be able to have any cognizance of things farther off, because no light waves can get across the boundary. Not that there is any real boundary: the people who live in what our observer takes to be lotus land live just as bustling a life as he does, but get the impression that he is eternally standing still. As a matter of fact, you would never become aware of the lotus land, because it would take an infinite time for light to travel from it to you. You could become aware of places just short of it, but it would remain itself always just beyond your ken. There will not be the ghostly suns of Einstein’s world, because light cannot travel so far.
One of the oddest things about this state of affairs is that empirical evidence for or against it is possible, and that there is actually some slight evidence in its favor. If all “clocks” are slowed down at a great distance from the observer, this will apply to the periodic motions of atoms, and therefore to the light which they emit. Consequently all rays of light emitted by distant objects ought, when they reach us, to look rather more red or less violet than when they started. This can be tested by the spectroscope. We can compare a known line, as it appears in the spectrum of a spiral nebula, with the same line as it appears in a terrestrial laboratory. We find, as a matter of fact, that in a large majority of spiral nebulæ there is a considerable displacement of spectral lines towards the red. The spiral nebulæ are the most distant objects we can see: Eddington states that their distances “may perhaps be of the order of a million light-years.” (A light-year is the distance light travels in a year.) The usual interpretation of a shifting of spectral lines towards the red is that it is a “Doppler effect,” due to the fact that the source of light is moving away from us. But one would expect to find the nebulæ just as often moving towards us as moving away from us, if nothing operated but the law of chances. If the world is such as De Sitter says it is, the spectral lines of the spiral nebulæ will be displaced towards the red owing to the slowing down of distant clocks, even if in fact they are not moving away from us. This, for what it is worth, is an argument in favor of De Sitter.
The same facts afford another argument in favor of De Sitter, for another reason. If, at a given moment, a body is at rest relatively to the observer, and at a distance from him, it will (in the absence of counteracting causes) not remain at rest from his point of view, but will begin to move away from him, and will continue to move away faster and faster; the further it is from him, the more its retreat will be accelerated. For bodies which are not too distant from each other, gravitation may overcome this tendency; but as this tendency increases with the distance, while gravitation diminishes, we should expect to find very distant bodies receding from us if De Sitter’s theory is right. Thus we have two reasons for the displacement of spectral lines in spiral nebulæ: one, the slowing down of time; the other, the movement away from us which we should expect at distances too great for gravitation to be sensible. However, it cannot be said that the argument, on either ground, is very strong. Eddington gives a list of forty-one spiral nebulæ, of which five have their spectral lines shifted towards the violet, not towards the red. Thus the material is neither very copious nor quite harmonious.
Einstein’s and De Sitter’s hypotheses do not exhaust the possibilities of a finite world: they are merely the two simplest forms of such a world. There are arguments against each, and it hardly seems probable that either is quite true. But it does seem probable that something more or less analogous is true. If the universe is finite, it is theoretically conceivable that there should be a complete inventory of it. We may be coming to the end of what physics can do in the way of stretching the imagination and systematizing the world. The period since Galileo has been essentially the period of physics, as the age of the Greeks was the period of geometry. It may be that physics will lose its attractions through success: if the fundamental laws of physics come to be fully known, adventurous and inquiring intellects will turn to other fields. This may alter profoundly the whole texture of human life, since our present absorption in machinery and industrialism is the reflection in the practical world of the theorist’s interest in physical laws. But such speculations are even more rash than those of De Sitter, and I do not wish to lay any stress upon them.
CHAPTER XII:
CONVENTIONS AND NATURAL LAWS
One of the most difficult matters in all controversy is to distinguish disputes about words from disputes about facts: it ought not to be difficult, but in practice it is. This is quite as true in physics as in other subjects. In the seventeenth century there was a terrific debate as to what “force” is; to us now, it was obviously a debate as to how the word “force” should be defined, but at the time it was thought to be much more. One of the purposes of the method of tensors, which is employed in the mathematics of relativity, is to eliminate what is purely verbal (in an extended sense) in physical laws. It is of course obvious that what depends on the choice of co-ordinates is “verbal” in the sense concerned. A man punting walks along the boat, but keeps a constant position with reference to the river bed so long as he does not pick up his pole. The Lilliputians might debate endlessly whether he is walking or standing still: the debate would be as to words, not as to facts. If we choose co-ordinates fixed relatively to the boat, he is walking; if we choose co-ordinates fixed relatively to the river bed, he is standing still. We want to express physical laws in such a way that it shall be obvious when we are expressing the same law by reference to two different systems of co-ordinates, so that we shall not be misled into supposing we have different laws when we only have one law in different words. This is accomplished by the method of tensors. Some laws which seem plausible in one language cannot be translated into another; these are impossible as laws of nature. The laws that can be translated into any co-ordinate language have certain characteristics: this is a substantial help in looking for such laws of nature as the theory of relativity can admit to be possible. Combined with what we know of the actual motions of bodies, it enables us to decide what must be the correct expression of the law of gravitation: logic and experience combine in equal proportions in obtaining this expression.